Suppose $A$ and $B$ are enemies. By the given condition, every other member cannot be a friend of both $A$ and $B$. Since each of $A$ and $B$ has 2 more enemies other than themselves, we must have $n \le 2 + 2 \times 2 = 6$. Also, as each member has 3 enemies, we have $n \ge 4$, and there are $\frac{3n}{2}$ pairs of enemies. This shows $n$ is even, and hence $n$ can only be 4 or 6.
To give examples for $n = 4$ and $n = 6$, we use terminologies in graph theory. Suppose vertices denote the members and edges join pairs of enemies. Then $K_4$ and $K_{3,3}$ are examples for $n = 4$ and $n = 6$ respectively.